You can
put this solution on YOUR website! Linear operators are not merely functions on sets,they are linear transfomrations(preserving addition and scalar multiplication of vectors on vector space.
I feel very strange that you haven't mentioned vector space, or matrices.
Without them, how can you start solving the questions ?
1) I'm looking for a linear operator from V->V such that T^2=0 but T does not =0.
Sol: Define
-->
generated by
T(1,0) = (0,1) and T(0,1) = (0,0)
[Note : (1,0) & (0,1) are standard basis of unitcolumn vectors]
Let B = {(1,0),(0,1)}
The matrix A = [T}B of T associated with the basis B as
[0 0]
[1 0]
or equivalently T(X) = AX for all column vector X in
Clearly,you can see that
but A is not 0.
More precisely, T(a, b) = aT(1,0) + bT(0,1) = a(0,1) = (0,a)
for all (a,b) in
Check: Clearly,T is not the zero operator (why?) and
we see that
(a,b) = T(0,a) = 0*T(1,0) = (0,0) for all real a,b
2) Given two linear operators (say, T, U) from V-> V, I'm looking for TU=0 but UT does not =0
Sol: Similarly to the example in 1)
Let T ,U be two linear operators in
defined by
T = (as matrix A)
[0 1]
[0 0] and
U = (as matrix B)
[1 0]
[0 0] then we have
TU =
[0 0]
[0 0] but UT =
[0 1]
[0 0] (not zero operator)
More precisely, define
-->
by T(X) = AX
and
-->
by T(X) = BX where X is any column vector of
.
AX & BX are products of matrices.
Make sure you do understand the examples above and work hard.
Kenny
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